Mean

The arithmetic average of the elements of the collection.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageMean!↗

Mean(collection)the arithmetic average of the elements.

SageMathmean(collection)Juliaenumeratio_mean(collection)

Mean(matrix)the column-wise mean, one value per column.

SageMathmean(matrix)Juliaenumeratio_mean(matrix)

Typed signatures

Type	Package	Applies
((collection<nothing | number> | distribution | number)+) -> number	compute-engine	always
((collection<nothing | number> | distribution | list<list<any>> | number)*) -> list | number	combinatoricsreplaces compute-engine	always
((collection<nothing | number> | distribution | list<list<any>> | number)*) -> list | number	statisticsreplaces combinatorics	always

Domain: Collections

Details

• $Mean(c) = \dfrac{\sum c}{Length(c)}$. See Length.
• Sensitive to outliers — a single extreme value can drag the mean far from the bulk of the data. See Median for a more robust alternative.
• Given a matrix (a list of equal-length rows), computes the mean of each column.
• See Mode for the most frequent value rather than the average.

Examples

Basic #

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In$\mathrm{mean}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)$Out$\frac{5}{2}$

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In$\mathrm{mean}(\bigl\lbrack2, 4, 4, 4, 5, 5, 7, 9\bigr\rbrack)$Out$5$

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In$\mathrm{mean}(\bigl\lbrack1.21, 3.4, 2.15, 4, 1.55\bigr\rbrack)$Out$2.462$

Scope #

Given a matrix, Mean is column-wise

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In$\mathrm{mean}(\bigl\lbrack\bigl\lbrack1, 10\bigr\rbrack, \bigl\lbrack2, 20\bigr\rbrack, \bigl\lbrack3, 30\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack2, 20\bigr\rbrack$

Exact rationals give an exact mean

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In$\mathrm{mean}(\bigl\lbrack\frac{1}{2}, \frac{1}{3}, \frac{1}{6}\bigr\rbrack)$Out$\frac{1}{3}$

Symbolic data gives an exact symbolic mean: $\frac{a+b+c+d}{4}$

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In$\mathrm{mean}(\bigl\lbrack a, b, c, d\bigr\rbrack)$Out$\frac{a+b+c+d}{4}$

Exact constants give an exact mean $\frac{2+e+\pi}{3}$, not a numeric approximation

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In$\mathrm{mean}(\bigl\lbrack π, e, 2\bigr\rbrack)$Out$\frac{2+e+π}{3}$

Of a matrix of symbols, the exact mean of each column

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In$\mathrm{mean}(\bigl\lbrack\bigl\lbrack a, u\bigr\rbrack, \bigl\lbrack b, v\bigr\rbrack, \bigl\lbrack c, w\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\frac{a+b+c}{3}, \frac{u+v+w}{3}\bigr\rbrack$

The mean of a lazy Range, exactly

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In$\mathrm{mean}(1..100)$Out$\frac{101}{2}$

The mean of a 30-digit numeric list is a float

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In$\mathrm{mean}(N(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, 30))$Out$2.5$

Stays symbolic for an unknown data list

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In$N(\mathrm{mean}(\mathrm{ts}))$Out$\mathrm{mean}(\mathrm{ts})$

The mean of the binomial distribution is $np$

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In$\mathrm{mean}(\operatorname{BinomialDistribution}(n, p))$Out$np$

For a custom density, the integral is not evaluated and the mean stays symbolic

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In$\mathrm{mean}(\mathrm{ProbabilityDistribution}(\frac{\sqrt{2}}{π((x-2)^4+1)}, \bigl\lbrack x, -∞, ∞\bigr\rbrack))$Out$\mathrm{mean}(\mathrm{ProbabilityDistribution}(\frac{\sqrt{2}}{π(x-2)^4+π}, \bigl\lbrack x, -∞, ∞\bigr\rbrack))$

Properties #

Mean is total divided by count: $Mean(c) = \dfrac{\sum c}{Length(c)}$. See Length

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In$\mathrm{mean}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)=\frac{1+2+3+4}{\mathrm{Length}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)}$Out$\mathrm{True}$

Numeric evaluation of the mean of eight values

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In$N(\mathrm{mean}(\bigl\lbrack1, 2, 3, 4, 4, 3, 1, 1\bigr\rbrack))$Out$2.375$

Elementwise functions apply before averaging

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In$\mathrm{mean}(\ln(\bigl\lbrack a, b, c, d\bigr\rbrack))$Out$\frac{1}{4}(\ln(a)+\ln(b)+\ln(c)+\ln(d))$

Central moments expand into the centred deviations

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In$\mathrm{mean}((\bigl\lbrack a, b, c\bigr\rbrack-\mathrm{mean}(\bigl\lbrack a, b, c\bigr\rbrack))^{n})$Out$\frac{1}{3}((-\frac{a}{3}-\frac{b}{3}+\frac{2c}{3})^{n}+(-\frac{a}{3}+\frac{2b}{3}-\frac{c}{3})^{n}+(\frac{2a}{3}-\frac{b}{3}-\frac{c}{3})^{n})$

The mean of $1..10$ is $11/2$

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In$\mathrm{mean}(1..10)$Out$\frac{11}{2}$

Possible issues #

A single outlier drags the mean far from the rest of the data, here to $-\frac{38}{3} \approx -12.67$. See Median for a more robust alternative

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In$\mathrm{mean}(\bigl\lbrack-100, 1, 1, 1, 1, 20\bigr\rbrack)$Out$-\frac{38}{3}$

The mean of no data is $NaN$

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In$\mathrm{mean}(\bigl\lbrack\bigr\rbrack)$Out$\operatorname{NaN}$

An outlier of $-100$ drags the mean far below the typical value

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In$N(\mathrm{mean}(\bigl\lbrack-100, 1, 1, 1, 1, 20\bigr\rbrack))$Out$-12.666\,666\,666\,666\,666\,666\,7$

About

Examples	21
Checked against	Wolfram Language 20 agree1 differSageMath 13 agree4 errorOscar 9 agreeJulia 10 agree1 error
Crosswalk	Wolfram LanguageSageMathJulia

Implementation

mappedoscarpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: Median, Mode